The rest of the paper is organised as follows.
In the next sections, we first recall the formal graph definitions (in Section ), the definition of the VRSP as well as the graph-decomposition theorems, together with other relevant terminology and notation (in Section ), the notions of graph isomorphism and contraction to labelled acyclic directed multigraphs (in Section ), and the two graph theorems given in [1]} (in Section ).
We relax the two theorems from [1]} and we use the VRSP and the two relaxed decomposition theorems to state and prove two decomposition theorems (in Section ).
